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Heat Kernels and Dirac Operators

  • Textbook
  • © 2004

Overview

Authors:
  1. Nicole Berline

    1. Centre de Mathématiques, École Polytechnique, Palaiseau CX, France

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  2. Ezra Getzler

    1. Department of Mathematics, Northwestern University, Evanston, USA

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  3. Michèle Vergne

    1. Centre de Mathématiques, École Polytechnique, Palaiseau CX, France

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Part of the book series: Grundlehren Text Editions (TEXTEDITIONS)

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About this book

The first edition of this book presented simple proofs of the Atiyah-Singer Index Theorem for Dirac operators on compact Riemannian manifolds and its generalizations (due to the authors and J.-M. Bismut), using an explicit geometric construction of the heat kernel of a generalized Dirac operator; the new edition makes this popular book available to students and researchers in an attractive softcover. The first four chapters could be used as the text for a graduate course on the applications of linear elliptic operators in differential geometry and the only prerequisites are a familiarity with basic differential geometry. The next four chapters discuss the equivariant index theorem, and include a useful introduction to equivariant differential forms. The last two chapters give a proof, in the spirit of the book, of Bismut's Local Family Index Theorem for Dirac operators.

Keywords

Table of contents (11 chapters)

  1. Front Matter

    Pages I-IX
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  2. Introduction

    • Nicole Berline, Ezra Getzler, Michèle Vergne
    Pages 1-12

  3. Background on Differential Geometry

    • Nicole Berline, Ezra Getzler, Michèle Vergne
    Pages 13-59

  4. Asymptotic Expansion of the Heat Kernel

    • Nicole Berline, Ezra Getzler, Michèle Vergne
    Pages 61-98

  5. Clifford Modules and Dirac Operators

    • Nicole Berline, Ezra Getzler, Michèle Vergne
    Pages 99-137

  6. Index Density of Dirac Operators

    • Nicole Berline, Ezra Getzler, Michèle Vergne
    Pages 139-161

  7. The Exponential Map and the Index Density

    • Nicole Berline, Ezra Getzler, Michèle Vergne
    Pages 163-179

  8. The Equivariant Index Theorem

    • Nicole Berline, Ezra Getzler, Michèle Vergne
    Pages 181-201

  9. Equivariant Differential Forms

    • Nicole Berline, Ezra Getzler, Michèle Vergne
    Pages 203-242

  10. The Kirillov Formula for the Equivariant Index

    • Nicole Berline, Ezra Getzler, Michèle Vergne
    Pages 243-262

  11. The Index Bundle

    • Nicole Berline, Ezra Getzler, Michèle Vergne
    Pages 263-310

  12. The Family Index Theorem

    • Nicole Berline, Ezra Getzler, Michèle Vergne
    Pages 311-348

  13. Back Matter

    Pages 349-363
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Authors and Affiliations

  • Centre de Mathématiques, École Polytechnique, Palaiseau CX, France

    Nicole Berline, Michèle Vergne

  • Department of Mathematics, Northwestern University, Evanston, USA

    Ezra Getzler

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